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On a pointwise inequality for even Legendre polynomials in high dimensional spheres

Shirong Chen, Yi C. Huang, Jian-Yang Zhang

Abstract

We present a pointwise inequality for adjacent even Legendre polynomials in high dimensional spheres featuring the effect of spectral gaps. This improves a recent result of Imbert, Silvestre and Villani that is crucially used in their study of the Fisher information for the Boltzmann equation.

On a pointwise inequality for even Legendre polynomials in high dimensional spheres

Abstract

We present a pointwise inequality for adjacent even Legendre polynomials in high dimensional spheres featuring the effect of spectral gaps. This improves a recent result of Imbert, Silvestre and Villani that is crucially used in their study of the Fisher information for the Boltzmann equation.

Paper Structure

This paper contains 2 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Proof of Proposition \ref{['p:LegPolGap']}

Key Result

Proposition 1

For the Legendre polynomials of order $\ell$ normalized so that $P_\ell(1)=1$, we have for all $x\in[-1,1]$ and $\ell\ge1$, Here $\lambda_\ell=\ell(\ell+d-2)$ is the eigenvalue of the (minus) spherical Laplacian $-\Delta_{\mathbb{S}^{d-1}}$ corresponding to spherical harmonics of order $\ell$, and the unique axially symmetric spherical harmonic $Y_\ell$ of order $\ell$ such that $Y_\ell(e_1)=1$ i

Theorems & Definitions (7)

  • Proposition 1: Imbert, Silvestre and Villani, 2024
  • Proposition 2
  • Remark 3
  • Remark 4
  • Lemma 5: Integral representation formula
  • Lemma 6
  • proof